1

Is there a reasonably standard symbol depending on a parameter, like $\delta_i$ or something, that takes the value $1$ when $i$ is even and $0$ when $i$ is odd? or the other way around?

$$ \frac{1 + (-1)^i}{2} $$ is far too cumbersome.

Edit: I really don't see why it was closed for lack of context. The context should be obvious -- the numerous formulas where such a symbol would be useful. If I asked, "is there a symbol depending on two parameters i and j which is 1 when i=j and 0 otherwise", I would have gotten the answer, "yes, the Kronecker delta symbol!" -- not a vote to put on hold.

But okay, if you want a particular context, I'll offer that this would give an alternative way to write coefficients of power series with alternating zero and nonzero terms.

There are other ways around that, so here's a different context. I'm writing a paper where I need to write a formula for the general term in the sequence of polynomials:

$x$, $-x+x^2$, $3x+x^2+x^3$, $-3x+4x^2-x^3+x^4$, $\ldots$

The coefficient of the $i$th degree term in the $m$th degree polynomial is, up to a sign,

$$\binom{\left\lfloor \frac{m+i-2}{2} \right\rfloor}{i-1}$$ then multiplied by $m/i$ when $m-i$ is even or $(m-1)/i$ when $m-i$ is odd. This latter factor would be easy to write using the symbol I asked about. I could also write this as a sum of two polynomials, one with odd degree terms and the other with even, but I would prefer one formula.

Barry Smith
  • 5,303
  • 2
    would (i+1)mod2 do? – Paul Apr 28 '15 at 12:45
  • Reason for down vote? I'm sure lots of people need such a symbol on occasion. What's wrong if asking if there is a standard? – Barry Smith Apr 28 '15 at 13:01
  • I don't like using mod as an operator, like the computer scientists do, and I know others who feel the same. But that's just personal aesthetics. Mostly, I hope there's something more compact. – Barry Smith Apr 28 '15 at 13:02
  • Okay, I can now see what you downvoters think the problem is, voting to close because it is off-topic. Seeing as there is a whole category on this site about mathematical notation, and I labeled the question as such, it is certainly an accepted topic at stack-exchange. I challenge you to point to one of the reasons in the "off-topic" list in the help center that applies here. You won't find one. This is NOT a question about typesetting. – Barry Smith Apr 28 '15 at 13:34
  • 2
    You can use Iverson brackets. – GFauxPas Apr 28 '15 at 14:05
  • @GFauxPas: Good idea in principle. In practice, I think it's rather optimistic to expect an unprepared reader to identify "$[2\mid i]$" as an Iverson bracket without an explicit explanation. – hmakholm left over Monica Apr 28 '15 at 14:28
  • So define it when you first mention it? Or make a footnote? – GFauxPas Apr 28 '15 at 14:41
  • Where are you using this, is it in a summation? – GFauxPas Apr 28 '15 at 14:49
  • I am already planning to define it before using it. I would just prefer to use a standard notation if there is one, hence my question. (I wouldn't use $\varepsilon_{ij}$ for the Kronecker delta, and I don't want to do something similar here.) – Barry Smith Apr 28 '15 at 14:56
  • Using $[i]$ to denote $i \pmod 2$, you could use the Kronecker delta: $\delta_{[i][1]}$. It's quite compact, though annoying to type... – pjs36 Apr 28 '15 at 14:57
  • @GFauxPas: I like modifying the Iverson bracket idea. I could write $[i]$ to mean $[i \equiv 0 \pmod{2}]$. I didn't know about those, so thanks. If this gets repoened and you make them an answer and nobody else comes up with something, I'll accept it. – Barry Smith Apr 28 '15 at 15:00
  • 2
    If $2{\mathbb N}$ is the set of even numbers, then you can use the characteristic function $1_{2{\mathbb N}}(x)$. It is quite common to use characteristic functions in analysis and very common in probability and statistics. – Taladris Apr 28 '15 at 15:06
  • @ Taladris. Thanks. My current application is more discrete math, so I'm leaning towards the brackets. But I'll keep the characteristic function in mind when I need to do something similar in the future. – Barry Smith Apr 28 '15 at 15:09
  • $$\mathbf 1_{i\ \text{even}}$$ – Did Apr 28 '15 at 15:42
  • Consider ${\rm odd}(x)$ which evaluates to ${\tt true}$ or $1$, when $x$ is odd, and similarly ${\rm even}(x)$. – Christian Blatter Apr 28 '15 at 15:59

1 Answers1

0

Consider the Iverson bracket convention.

If $p$ is a statement: $[p] = \begin{cases} 1; & p \\ 0; & \neg p\end{cases}$

If $P$ is a predicate: $[P(x)] = \begin{cases} 1; & P(x) \\ 0; & \neg P(x)\end{cases}$

So your desired notation would be $[2 | i]$ or $[i \equiv 0 \bmod 2]$

Or maybe (these two I made up):

$[\mathcal{E}(i)]$, or $[i \equiv 0]_2$

Or the characteristic function on evens:

$$\chi_{\mathcal E}(i) = \mathbf{1}_{\mathcal E}(i)$$

I use $\mathcal E$ and $\mathcal O$ for the set of even and odd numbers in my personal notes, but of course you can use some other symbol.

edit: I've seen $[p]_q$ to mean the remainder of $p / q$, I think, somewhere. So that would be $[i]_2$.

GFauxPas
  • 5,047